$18\mathrm{.}1$ Three dissimilar cylindrical rods are jointed together as indicated in Fig. $18\mathrm{.}14.$

The brass cylinder is fixed to the rigid support at the top and the aluminium

Fig. $18\mathrm{.}14$

cylinder may be regarded as simply$-$supported with a point load at mid$-$span.

Using the stiffness matrix approach, calculate the movement at each interface and

the forces at the rigid supports when $25$ kN and $100$ kN forces are applied as

indicated. $E_b=100G\frac{N}{m^2},E_s=200G\frac{N}{m^2},E_a=70G\frac{N}{m^2}.$

Example $10\mathrm{.}3$

A rectangular bar with shoulder fillet is subjected to a uniform bending moment of

$100$ Nm $.$ Its dimensions are as follows $($see Fig. $10\mathrm{.}22)D=50mm$;$d=25mm$;$r=$

$2\mathrm{.}5mm$;$h=10mm.$

Determine the maximum stress present in the bar for static load conditions. How would

the value change if $($a$)$ the moment were replaced by a tensile load of $20$ kN $,($b$)$ the moment

and the tensile load are applied together.

Example $10\mathrm{.}4$

A semi$-$circular groove of radius $3$ mm is machined in a $50$ mm diameter shaft which is

then subjected to the following combined loading system:

$($a$)$ a direct tensile load of $50$ kN $,$

$($b$)$ a bending moment of $150$ Nm $,$

$($c$)$ a torque of $320$ Nm $.$

Determine the maximum value of the stress produced by each loading separately and hence

estimate the likely maximum stress value under the combined loading.